Almost every bike that was check for wheel alignment at the Heart of Washington Ride was off. Mine was 1 3/8 toed in on the right wheel and a smidge towed on the left. when I went around a right hand corner it would over steer but a small amount of correction would cause it to over correct and pull me into the turn. Left hand turns were less obvious. I also noticed that it would dance left to right if I was not firm on the grips or shifted my weight, especially while standing to get butt relief.
After the alignment she track straight and corners perfectly now, with or with out the trailer. I tested it to 100 mph by GPS and she would track straight without hands.
I would consider a laser alignment before anything else but have the bearings and tie rod ends inspected while they are at it. If it is as bad as you say I doubt shocks setting or tire pressures are to blame.
Good luck with it and ride safe